Dynamical systems defined on networks have applications in many fields in science and engineering. In particular, it is important to understand when networks exhibit synchronous or other types of coherent collective behaviors. Other questions include whether such coherent behavior is stable with respect to random perturbation, or what the detailed structure of this behavior is as it evolves. We will examine several models of networked dynamical systems and present a mixture of results that range from rigorous theorems for abstract models to quantitative comparisons of models and data.